Patterns and Algebra - The Cartesian Coordinate Plane (Four Quadrants)

The Cartesian Plane is a two-dimensional grid formed by the intersection of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They meet at a central point called the origin, which is labeled $(0, 0)$. This grid allows us to locate any point in space using a set of coordinates.

An Ordered Pair $(x, y)$ specifies the exact position of a point. The first value, the x-coordinate, represents the horizontal distance from the origin (move right for positive values, left for negative values). The second value, the y-coordinate, represents the vertical distance (move up for positive values, down for negative values).

The intersection of the axes divides the plane into four distinct regions called Quadrants, numbered I to IV in a counter-clockwise direction starting from the top-right. Quadrant I (top-right) contains points with signs $(+, +)$, Quadrant II (top-left) contains $(-, +)$, Quadrant III (bottom-left) contains $(-, -)$, and Quadrant IV (bottom-right) contains $(+, -)$. Visually, this creates a cross shape with the origin at the center.

To plot a point such as $(-2, 3)$, you begin at the origin $(0, 0)$. First, move $2$ units to the left along the x-axis because the x-coordinate is negative. Then, move $3$ units vertically up because the y-coordinate is positive. Mark the point where these two paths meet.

Points located directly on the axes always have at least one coordinate equal to zero. If a point is on the x-axis, its y-coordinate is zero, such as $(5, 0)$. If a point is on the y-axis, its x-coordinate is zero, such as $(0, -4)$. These points do not belong to any specific quadrant but lie on the boundary between them.

Reflections are transformations where a point is 'flipped' over an axis. When reflecting point $(x, y)$ across the x-axis, the x-coordinate stays the same but the y-coordinate changes sign, resulting in $(x, -y)$. When reflecting across the y-axis, the y-coordinate stays the same but the x-coordinate changes sign, resulting in $(-x, y)$. This creates a mirror image across the chosen axis line.

Horizontal and Vertical distance can be calculated by finding the absolute difference between coordinates. If two points have the same y-coordinate, the distance between them is the absolute difference of their x-coordinates. If they have the same x-coordinate, the distance is the absolute difference of their y-coordinates.